Mathematical Model in Studying the Stability of Dynamic Systems

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Mathematical Model in Studying the Stability

of Dynamic Systems

Le Hong Lan

Department of Basic Science, Transport and Communications University, 
No.3, Cau Giay, Hanoi, Vietnam

Received 25 April 2017

Revised 30 May 2017; Accepted 05 June 2017

Abstract: In this paper, the author investigated the phenomenon of flutter, which may be the cause 
of instability of construction structure when it is affected by aerodynamics. By analyzing the effect
of aerodynamic on the structure via mathematical analysis, the author has established a
mathematical model to study the stability of the structure in the aerodynamic flux that moves
supersonically.

Keywords: Aerodynamics, flutter, stability.

1. Introduction

Aerodynamic load is a significant element when calculating physical structure of large
constructions, such as skyscrapers, antenna towers, suspension bridges, etc. This is especially
important for flying equipments. Aerodynamic that affects on the structures in both the same direction
as well as the perpendicular one of gas flow, depends on magnitudes, characteristics of the gas flow
and the movements of the structures. Aerodynamic stability of technical systems is the most important
consideration when calculating structures under aerodynamic load. The study of static and dynamic
nonlinear patterns of flutter and the aerodynamic stability has a significant impact in practical problems.

Although there are many works dealt with the stability of sheet structures which have composite
materials, however, almost the previous publications, the authors have mainly applied finite element
methods [1-8]. The main content of this paper is a dynamic analysis of sheet structure that has
functionally graded materials under aerodynamic load. One of the unstable situations is flutter, which
might cause the instability of structure under aerodynamic load. In the sheet structure the energy from
gas flow was generated depending on the movement and speed of such movement. By using nonlinear
Piston theory and mathematical analysis, we proposed a qualitative model of nonlinear flutter of
structures, and evaluated the stability of the model by numerical analysis.

_______



Corresponding author. Tel.: 84-989060885.
Email:

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2. Content

2.1. The model of nonlinear flutter of sheet structure under aerodynamic load

Let us investigate a sheet structure from ceramic and metal with the depth h , the length a and the 
width b. The outside of the sheet structure is effected by the supersonic gas flow in the parallel 
direction with the middle surface of the structure. In the coordinate axis 0 xyz, axes 0x, 0 y and

0 z describe different directions of the middle surface of the sheet structure.

The ratio between the volume of ceramic V ( z ) and the one of metal c V ( z ) is distributed m

according to the mixed law

k

c m c

2z h
V ( z ) V ( z ) 1, V ( z ) ,

2h


 

    

  (1)
in which, k0 is the volume-fraction index, z is the thickness coordinate of the sheet structure,

h h

z ,

2 2

 

  

 . According to the mixed law, module Young E z

 

and mass density 

 

z of the
material are expressed under the form

k

c c m m m c m

k

c c m m m c m

2z h
E( z ) E V E V E ( E E )

2h
2z h

( z ) V V ( ) ,

2h

     

   

    

  

  





 

     

 

  



(2)

where E ,c E and m c

 

z , m

 

z are respectively the module Young and mass densities of the
ceramic and of the metal.

The movement of a material point M x, y,z





in the sheet structure has moving components
u, v and w in the directions of 0x, 0 y and 0z. The transpose in a neighborhood of the point consists 
of transposed parts that cause the strains and circular motion. The distorted elements are constrained
by the conditions that are suitable for the change, in order to ensure the existence of continuous and
monotonic solutions.

According to the classical theory of sheet structure with nonlinear geometrical characteristics,

so-called Von Korman-Donnell, deformations 

 

M 



  x, y, xy



at the point M x, y,z having the





z

0

x

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distance z from the middle surface of the sheet structure are presented by transposed elements 
u, v, w by deformation 0 



  x0, y0 , 0xy



and by vector curvature of bending sheet 



  x, y, xy



via the relation 

 

M 0z. Consequently, we have

x x0z x, y 0y zy, xy0xy2zxy, (3)
where

2 2

0

x x 2

2 2

0

y y 2

2
0

xy xy

u 1 w w

,

x 2 x x

v 1 w w

,

y 2 y y

u v w w w

, .

y x x y x y

 
 
 
    
  
    


    
   
    
 

     
    
     

(4)

Then, the deformation compatibility equation is

2

2 0 2 0

2 0 2 2 2

y xy

x

2 2 2 2

w w w

.

y x x y x y x y

 

    

   

    

         (5)
Hooke’s law describes the relationship between stress and deformation of the structure as follows,
with the Poisson's ratio v is assumed to be constant

x E( z )2 ( x y), y E( z )2 ( y x), xy E( z ) xy.

1 1 2( 1 )

         

  

    

   (6)
Components of internal force and moment are calculated through stress components





h
2
0 0
1 2

x x 2 x y 2 x y

h
2
h
2
0 0
1 2

y y 2 y x 2 y x

h
2
h
2
0
1 2

xy xy xy xy

h
2

E E

N dz ( ) ( )

1 1

E E

N dz ( ) ( )

1 1

E E

N dz ,

2 1 1

      
 
      
 
  
 




     
  




     
  




  
  




(7)

 
h
2

0 0 3

2

x x 2 x y 2 x y

h
2
h
2

0 0 3

2

y y 2 y x 2 y x

h
2
h
2
0 3
2

xy xy xy xy

h
2

E
E

M z dz ( ) ( )

1 1

E
E

M z dz ( ) ( )

1 1

E
E

M z dz ,

2 1 1

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in which,

 







 



 



h
2
c m
1 m
h
2
h
2
2
c m
2
h
2
h
2

2 m 3

3 c m

h
2

E E

E E z dz E h

k 1

E E

E E z z dz k h

2 k 1 k 2

E 1 1 1

E E z z dz E E h .

12 k 3 k 2 4 k 1







 
   
 
   




  

 



   
       
   
   
 
   
 




(9)

From (7) it follows that









0 2

x x y x

1 1

0 2

y y x x

1 1

0 2

xy xy xy

1 1
E
1
N N
E E
E
1
N N
E E

E
1

2 N .

E E
  
  

 

   

   


   
    
  

(10)

Combining with (8) we obtain













2

1 3 2

2

x x 2 x y

1 1

2

1 3 2

2

y y 2 y x

1 1

2

1 3 2

2

xy xy 2 xy

1 1

E E E

E

M N ( )

E E 1

E E E

E

M N ( )

E E 1

E E E

E

M N .

E E 1

  

  




 
   




   





  
 

(11)

To establish mathematical equations of a sheet structure under aerodynamic load, we consider the
case that a sheet lies in the same direction with the movement direction of the supersonic gas flow U .

The flow affects on the surface of the sheet structure via the pressure q in the perpendicular 0
direction with the middle layer of the sheet. According to Love’s theory, the equations of motion are
given by

2
xy
x
0 2

2
xy y
0 2
N
N u

x y t

N N v

x y t




    
   

   
  
   

(12)
2 2
2 2
xy y
x

x xy xy y 0 0

2 2 2

M M

M w w w w w

2 N N N N q ,

x x y y x x y y x y  t

     

       

         

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where

h / 2

c m

0 m

h / 2

( z )dz h.

k 1

 

  



 

   



 



According to the

nonlinear

Piston theory, aerodynamic force affecting on the structure is defined
by the formula

q0 P M w 1 w ,

x a t

 



   

    

 

  (14)
in which  is the heat capacity of gas, a is the speed of sound, P is the gas pressure without
perturbation. With U is the speed of gas flow, M : U

a

 is the Mach number, which characterizes
the compression strength of the moving gas flow.

Using Volmir’s hypothesis, we have

2 2

0 2 0 2

u v

0, 0

t t

     

  as u w, v w . By choosing a 
stress function  such that

2 2 2

x 2 y 2 xy

N , N , N ,

y x x y

  

  

   

    (15)
the equations of (12) becomes homogeneous equations.

The equation (13) is transformed into

2 2

2 2 2 2 2

xy y

x

x xy y 0 0

2 2 2 2 2

M M

M w w w w

2 N 2N N q .

x x y y x x y y  t

 

           

         (16)

From (5), (10) and (15) we transform the deformation compatibility equation into

2

2 2 2

2 2

1

1 w w w

.

E  x y x y

    

  

   

  (17)
From (11), (14), (15) and (16) we have the movement equation





2

2 2 2 2 2 2 2

1 3 2

0 2 2 2 2 2 2

1

E E E

w w 1 w w w w

P M w 2 0.

t x a t E 1 x y x y y x x y

  

  





  

        

       

              (18)

The system of equations (17) and (18) has unknown functions  and w describing the flutter of 
the structure under aerodynamic force. This system is used to study the nonlinear flutter and dynamic
stability of structure.

2.2. The analysis of nonlinear dynamic models

Consider the single mixed sheet in each edge, that satisfies the boundary conditions





 





 

x y xy x 0 ,x a

y y xy y 0 , y b

w M N N 0

w M N N 0.

 

    




   




(19)

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1 2

m x n y ( m 1 ) x n y

w f ( t )sin sin f ( t )sin sin ,

a b a b

    

 

(20)
Such that, they satisfy the boundary condition (19) and the initial condition:





 0





 0





 0





 0

1 1 1 11 2 2 2 21

f t0  f , f t0  f , f t0  f , f t0  f . (21)
Solving the equation (17) with the formula of the equation’s solution determined by (20), we find a
stress function  in the form

1 2 3 4

5 6 7

2n y 2m x 2( m 1 ) x ( 2m 1 ) x

cos cos cos cos

b a a a

x 2n y ( 2m 1 ) x 2n y x

cos cos cos cos cos ,

a b a b a

   
    
    
  
 
   

  
(22)
where

























2

2 2 2

2 2 2 2 2 2 2 2

1 1 2

1 1 1 2 1 1 2

1 2 2 2 2 3 2 4 2

2

2 2

2 2 2 2

1 1 2

1 1 2 1 1 2

5 6 2 2 7 2 2 2

2 2

E f m f m 1 E f n E f n E f f n

, , ,

32n 32m 32 m 1 4 2m 1

E f f n 2m 1

E f f n E f f n

, , .

4 4 2m 1 4n 4 1 4n

  
   


 
  


   
 
   
 

 
  
    
 
(23)

Using the Galerkin’s method for the equation (18), we obtain the system







































2 4 2 4 4

4 2

2 2 2

1 3 2 1 1 2 1

0 4 1 2 1 4 2

1

2 4

2 3

4 4 4

1 1

2 2

4 4 2 2 2

2 2

2

4 2

2 2 2

1 3 2

0 4 2 2 2

1

4Mm m 1

E E E P a f f f m E

a

f m n f f

a 2m 1 a 16

E 1

m m 1 2m 1 1 f E

4 m n 0

n 1 4n 2m 1 4n 16

E E E

a

f m 1 n f

E 1
 
 
  

  

 
 


  

     

  
 
 

 
      
 
  
   
 
  
     






























4 2 4 4

2 2 1 1

1
4

2 4

2 3

4 4 4

2 1

2 2

4 4 2 2 2

2 2

4Mm m 1

P a f f f n E

f

a 2m 1 a 16

m m 1 2m 1 1 f E

4 m 1 n 0.

n 1 4n 2m 1 4n 16




  












  
  
 
   
 

 
    
 
         
       
 


(24)

Then, we have a relation that determines the partial flutter frequency  of the sheet structure as below

2

K H 0, (25)
with



























3
2
2

2 2 2

1 3 2

4
2

1

3 2 2

2 2 2

1 3 2

4 2

1

4 P a Mm m 1

E E E

m n

2m 1

E 1

K

4 P a Mm m 1 E E E

m 1 n

2m 1 E 1







 


    

  

 
   

     
 
   
 

;
4
0 4
4
0 4
a
0
H .
a
0




 
 
 

 
 
 

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2.3. Fitting of the model

To evaluate the reliability of the method, we study the influence of composite elements of a
material on the stability of flutter frequency by using numerical analysis.

The system of nonlinear differential equations (24) satisfies the initial condition

 

10

 

10

 

1 1 2 2

f 0 10 , f 0 0, f 0 10 , f 0 0.

Considering the case where geometrical parameters of the structure are assigned values: Ec 
380.109 N/m2, c  3800 kg/m3, Em70. 109 N/m2, m  2720 kg/m3;  0,3 and the characterized
factors of aerodynamic force:   1,4, P  99473,4 N/m2, a 340m/s.

The amplitude of oscillation f t , f1

   

2 t of sheet structure, that are described as in the Figures 1, 
2, 3, 4below shows that the mathematical model is suitable to describe the influence of the component
elements.

Figure 1. Stable flutter Figure 2. Instable flutter when
sheet depth reduces 0,001m.

With the data described in Figure 1 and Figure 2, we observed that with a small change of the depth
of the sheet structure, the flutter will change from a stable state to an unstable one.

Figure 3. Stable flutter Figure 4. Instable flutter when increasing the ratio a/b to 0.01116.

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3. Conclusion

The mentioned above mathematical model can be used to study the qualitative problems of
technical systems of sheet structure of constructions under aerodynamic force. The application of
dynamic criteria in investigating the stability of mechanical systems leads to the investigation of the
stability of solution of differential equations that describes the motion.

References

[1] F.Sabri, A.A. Lakis, (2013) Efficient Hybrid Finite Element Method for Flutter Prediction of Functionally Graded
Cylindrical Shells, Journal of Vibration and Acoustics 136. First published as doi: 10.1115/1.4025397.

[2] McNamara J.J., Friedmann P.P., Powell K.G., Thuruthimattam B.J. (2005), “ Three- dimensional Aeroelastic and
Aerothermoelastic behavior in Hypersonic Flow”, 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural
Dynamics & Material Conference 18-21 April 2005, Texas.

[3] Barbero E. J, Reddy J. N. (1990). “Nonlinear analysic of composite laminated plates using a generalized laminate
plate theory”, AIAA Journal, Vol. 28, No.1, pp 1987-1994.

[4] Beldica C. E., Hilton H.H. and Kubair D. (2001). “Viscoelastic panel flutter-stability, probabilities of failure and
survival times”, Submitted to The 42nd

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics &
Material Conference 16-19 April 2001, Seattle, W.A.

[5] Dey P. and Singha M. K. (2006). “Dynamic stability analysis of composite skew plates subjected to periodic in-
plane load”, Thin-Walled Structures 44, pp. 937 - 942.

[6] Singha M. K and Ganapathi M. (2005). “ A parametric study on supersonic flutter behaviour of laminated
composite skew flat panels”, Composite Structures 69, pp. 55-63.

[7] Long N. V., Quoc T. H., Tu T. M. (2016) “Bending and free vibration analysis of functionally graded plates using
new eight- unknown shear deformation theory by finite element method”, Journal of Science and Technology 54
(3), 402-415.

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